EDEXCEL GCSE MATHS · HIGHER
Exam Practice: Solving problems in 3D
Solving problems in 3D · More trigonometry · Lesson 7 of 9 · 19 marks · 30 minutes
|
Name |
|
Date |
|
Instructions
• Answer all the questions.
• Write your answers in the spaces provided.
• The marks for each question are shown in brackets - use this as a guide to how much to write.
• The answers are on separate pages at the back. Attempt every question before you look at them.
• Answer all questions. Show your working.
Question 1 WORK OUT (3 marks)
A cuboid is 12 cm long, 4 cm wide and 3 cm high. Work out the length of the longest diagonal of the cuboid.
|
|
|
|
|
|
|
|
(Total for Question 1 = 3 marks)
Question 2 WORK OUT (4 marks)
ABCDEFGH is a cuboid with AB = 8 cm, BC = 6 cm and CG = 5 cm. Work out the size of the angle between AG and the base ABCD. Give your answer correct to 1 decimal place.
|
|
|
|
|
|
|
|
|
|
|
|
(Total for Question 2 = 4 marks)
Question 3 WORK OUT (3 marks)
The diagram shows a pyramid with a square base of side 8 cm. The vertical height of the pyramid is 10 cm and the apex V is directly above the centre of the base. Work out the length of the sloping edge VA. Give your answer correct to 3 significant figures.
|
|
|
|
|
|
|
|
(Total for Question 3 = 3 marks)
Question 4 WORK OUT (4 marks)
A pyramid has a square base of side 8 cm and a vertical height of 10 cm. The apex is directly above the centre of the base. Work out the angle between a sloping edge and the base. Give your answer correct to 1 decimal place.
|
|
|
|
|
|
|
|
|
|
|
|
(Total for Question 4 = 4 marks)
Question 5 WORK OUT (3 marks)
AB is a vertical pole, 9 m tall, standing on level ground at B. Points C and D are on the ground with BC = 12 m, CD = 5 m and angle BCD = 90°. Work out the angle of elevation of A from D. Give your answer correct to 1 decimal place.
|
|
|
|
|
|
|
|
(Total for Question 5 = 3 marks)
Question 6 EXPLAIN (2 marks)
Explain how to find the angle between a line and a plane.
|
|
|
|
|
|
(Total for Question 6 = 2 marks)
TOTAL FOR PAPER = 19 MARKS
|
|
Answers and mark scheme
Check your answer only once you have written one.
Question 1 (3 marks)
√(12² + 4² + 3²) = √169 = 13 cm
• 12² + 4² + 3² M1
• 169 A1
• 13 A1
Question 2 (4 marks)
AC = √(8² + 6²) = 10; tan θ= 5/10, so θ= 26.6°
• AC = √(8² + 6²) M1
• AC = 10 A1
• tan θ= 5/10 M1
• 26.6 A1
Question 3 (3 marks)
Half the base diagonal = √(8² + 8²)/2 = 5.657; VA = √(10² + 5.657²) = 11.5 cm
• Half diagonal 5.657 M1
• 10² + 5.657² M1
• 11.5 A1
Question 4 (4 marks)
tan θ= 10/5.657, so θ= 60.5°
• Half diagonal 5.657 M1
• tan θ= 10/5.657 M1
• Correct angle expression A1
• 60.5 A1
Question 5 (3 marks)
BD = √(12² + 5²) = 13; tan θ= 9/13; θ= 34.7°
• BD = 13 M1
• tan θ= 9/13 M1
• 34.7 A1
Question 6 (2 marks)
Draw the perpendicular from one end of the line down to the plane. The angle between the line and its projection on the plane (in the right-angled triangle formed) is the angle required.
• Perpendicular to the plane M1
• Angle in the right-angled triangle C1